Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
セミパラメトリック推論の基礎の復習
Search
Daisuke Yoneoka
November 14, 2023
Research
0
16
セミパラメトリック推論の基礎の復習
Daisuke Yoneoka
November 14, 2023
Tweet
Share
More Decks by Daisuke Yoneoka
See All by Daisuke Yoneoka
感染症の数理モデル4
kingqwert
0
9
感染症の数理モデル3
kingqwert
0
38
感染症の数理モデル2
kingqwert
0
42
感染症の数理モデル1
kingqwert
0
69
媒介分析と疫学
kingqwert
0
120
時系列解析と疫学
kingqwert
2
940
Supervised PCAとその周辺
kingqwert
1
24
Sparse linear models
kingqwert
0
14
SKAT for rare common variants
kingqwert
0
12
Other Decks in Research
See All in Research
[KDD2023論文読み会] BERT4CTR: An Efficient Framework to Combine Pre-trained Language Model with Non-textual Features for CTR Prediction / KDD2023 LY Tech Reading
shunk031
0
470
VAR モデルによる OSS プロジェクト同士が生存性に与える 影響の分析
noppoman
0
140
How to Perform Manual Classification for Deep Learning Using CloudCompare
kentaitakura
0
670
プロシェアリング白書2024_PROSHARING_REPORT_2024
circulation
0
710
研究効率化Tips_2024 / Research Efficiency Tips 2024
ryo_nakamura
5
3.1k
第12回全日本コンピュータビジョン勉強会:画像の自己教師あり学習における大規模データセット
naok615
0
530
[Human-AI Decision Making勉強会] 説明の更新はユーザにどのような影響をもたらすか
okoso
1
210
LLMマルチエージェントを俯瞰する
masatoto
26
16k
サウナでのプロジェクションマッピングの可能性の検討 / EC71koizumi
yumulab
0
190
Accurate Method and Variable Tracking in Commit History
tsantalis
0
280
20240127_熊本から今いちど真面目に都市交通~めざせ「車1割削減、渋滞半減、公共交通2倍」~ 全国路面電車サミット2024宇都宮
trafficbrain
1
680
Generative AI - practice and theory
gpeyre
1
590
Featured
See All Featured
Cheating the UX When There Is Nothing More to Optimize - PixelPioneers
stephaniewalter
275
13k
Principles of Awesome APIs and How to Build Them.
keavy
121
16k
What's new in Ruby 2.0
geeforr
337
31k
Producing Creativity
orderedlist
PRO
338
39k
The Language of Interfaces
destraynor
151
23k
GraphQLの誤解/rethinking-graphql
sonatard
56
9.3k
Building Effective Engineering Teams - LeadDev
addyosmani
32
1.9k
GraphQLとの向き合い方2022年版
quramy
33
12k
Distributed Sagas: A Protocol for Coordinating Microservices
caitiem20
323
20k
Designing for Performance
lara
601
67k
4 Signs Your Business is Dying
shpigford
176
21k
Done Done
chrislema
178
15k
Transcript
ηϛύϥϝτϦοΫਪͷجૅͷ෮श Daisuke Yoneoka September 29, 2014
Notations جຊతʹ Tsiatis,2006 ʹै͏. Θ͔Μͳ͔ͬͨΒࣗͰௐͯͶ! ϕΫτϧߦྻଠࣈʹͯ͠ͳ͍͚Ͳ, ͦࣗ͜Ͱิ͍ͬͯͩ͘͞. σʔλ i.i.d Ͱ
Zi = (Zi1, . . . , Zim) ∈ Rm αϯϓϧαΠζ n ਓ. i.e., Z1, . . . , Zn φ(Z) Өڹؔ u(Zi, θ) ਪఆؔ Լ͖ࣈͷ eff (ۙ) ༗ޮ (efficient) ͱ͍͏ҙຯ
ηϛύϥϝτϦοΫਪͱʁ Zi ͷີ͕ؔηϛύϥϝτϦοΫϞσϧʹै͏ͱ S = {p(z : θ, η)|θ ∈
Θ ⊂ Rr, η ∈ H} θ ༗ݶ࣍ݩͷڵຯ͋ΔύϥϝλͰ, η ແݶ࣍ݩͷͲ͏Ͱ͍͍ύ ϥϝλ (ہ֎ (nuisance) ύϥϝʔλʔ). ηϛύϥϝτϦοΫਪ: ͜ͷͱͰ θ ͷ࠷ྑͷਪఆྔ (RAL ਪఆ ྔ) ΛͱΊΔ͜ͱ
Өڹؔ θ ͳΜͰ͍͍͔Β࠷ྑΛݟ͚ͭΔͱ͍͏ͷແཧήʔ → Ϋϥε Λݶఆͯͦ͜͠Ͱݟ͚ͭΔ! (౷ܭͰΑ͘ΔΑͶ) Өڹؔ: ਪఆྔ ˆ
θ ͷӨڹؔͱ, (Ϟʔϝϯτʹ੍͕͋Δ) √ n(ˆ θ − θ) = 1 √ n n i=1 φ(Zi, θ, η) + op(1) Λຬͨ͢ϕΫτϧؔ. ˆ θ ۙઢܗਪఆྔͱݺͼ n → ∞ ͰҰகੑ ͱۙਖ਼نੑ͕͋Δ √ n(ˆ θ − θ) → N 0, E[φ(Zi, θ, η)φ(Zi, θ, η)T ] Πϝʔδతʹ͋Δσʔλ͕ͲΕ͚ͩਪఆʹӨڹΛ༩͍͑ͯΔ͔Λ දݱͨ͠ͷ
ਪఆؔͱ M ਪఆ ਪఆํఔࣜ n i=1 u(Zi, θ) ਪఆؔ =
0 ͷղͱͯ͠ಘΒΕΔͷΛ M ਪఆྔ ͱݺͿ. Α͘ݟΔ score ؔͳΜ͔ίϨ. ͨͩ͠, E[φ(Zi, θ)] = 0 ظ 0 , E[∥φ(Zi, θ)∥2] < ∞ ࢄతͳͷൃࢄ͠ͳ͍ . ͋ͱ͏গ͚ͩ݅͋͠Δ. Ұகੑͱۙਖ਼نੑΛ࣋ͭ √ n(ˆ θ − θ) = 1 √ n n i=1 E[ ∂u(Zi, θ) ∂θ ] −1 u(Zi, θ) ͕͜͜Өڹؔʹͳ͍ͬͯΔ +op(1) → N 0, E[ ∂u(Zi, θ) ∂θ ] −1 E[u(Zi, θ)u(Zi, θ)T ] E[ ∂u(Zi, θ) ∂θ ] −T ] ͜ͷۙࢄͷਪఆྔΛαϯυΠονਪఆྔͱݺΜͩΓ͢Δ
RAL ਪఆྔ ۙઢܥਪఆྔͳΜ͔ྑͦ͞͏ʂͰ super efficiency ͷ (Hodges) ͕Δʂ Super efficiency:
ۙతʹ Cramer-Rao ͷԼݶΑΓྑ͍ͷ͕Ͱ͖ Δͷ͜ͱ ͜ͷΛղܾͨ͠ͷ͕ RAL (Regular asymptotic linear) ਪఆྔ. ͦͷਖ਼ଇ݅ۃݶ͕ LDGP (local data generating process) ʹґ ଘ͠ͳ͍͜ͱ (ৄ͘͠ Tsiatis, 2006) ηϛύϥਪ͜ͷ RAL ਪఆྔͷӨڹؔΛٻΊΔ͜ͱΛߟ͑Δ
Parametric submodel ηϛύϥϝτϦοΫϞσϧ S ͷ֤ʹର͠ p(z; θ, η) ∈ Ssub
⊂ S Λຬͨ͢ύϥϝτϦοΫϞσϧ Ssub = {p(z; θ, γ)|θ ∈ Θ ⊂ Rr, γ ∈ Γ ⊂ Rs, s ∈ N} ΛύϥϝτϦοΫαϒϞσϧͱݺͿ.
Nuisance tangent space (ہ֎ۭؒ) ηϛύϥϝτϦοΫϞσϧ S ͷ֤ʹର͠, ύϥϝτϦοΫαϒϞσϧ Ssub ͷہ֎ۭؒΛ
TN θ,γ (Ssub) = {BT sγ(z, θ, γ)|B ∈ Rs} ͱ͢Δ. γ p(z; θ, η) ʹରԠ͢ΔͷͰ sγ(z, θ, γ) = ∂ ∂γ log p(z; θ, γ) Ͱ ද͞ΕΔ nuisance score ؔ. ͜ͷઢܗۭؒ͜ͷ nuisance score vector ʹ ΑͬͯுΒΕ͍ͯΔ. ͜ͷͱ͖ TN θ,η (S) = Ssub TN θ,γ (Ssub) Λ S ্ͷ p(z; θ, η) ʹ͓͚Δہ֎ۭؒͱΑͿ. ͪͳΈʹ, ଆͷू ߹ʹؔͯ͠ closure ΛͱΔԋࢉࢠ. Note:͜ͷۭؒେͰޙʹ, RAL ਪఆྔͷӨڹؔ͜ͷۭؒʹަۭͨؒ͠ʹ ଐ͢Δ͜ͱ͕ॏཁʹͳͬͯ͘Δʂ
ઢܗ෦ۭؒͷࣹӨͷزԿͱϐλΰϥεͷఆཧ
RAL ਪఆྔͷӨڹؔͷॏཁͳఆཧ ηϛύϥϝτϦοΫ RAL ਪఆྔ β ͷӨڹؔ φ(Z) ҎԼͷ݅Λຬ ͢Δ.
Corollary1 E[φ(Z)sβ] = E[φ(Z)sT efficient (Z, β0, η0)] = I. ͨͩ͠, s είΞؔͰ, sT efficient ༗ޮείΞؔ Corollary2 φ(Z) ہ֎ۭؒʹަ͍ͯ͠Δ. ༗ޮӨڹ্ؔͷ 2 ͭͷ݅Λຬͨ͠, ͦͷࢄߦྻ, ޮݶքΛୡ ͦ͠Ε φeffi(Z, β0, η0) = E[seff (Z, β0, η0)sT eff (Z, β0, η0)] −1 seff (Z, β0, η0)
ηϛύϥۭؒͷఆཧ ύϥϝτϦοΫαϒϞσϧͷ߹ͷ RAL ਪఆྔͷӨڹؔͱۭؒͱͷؔ Tsiatis, 2006 ͷ Ch4.3 ͋ͨΓΛݟͯͶʂ ఆཧ
1 RAL ਪఆྔͷӨڹؔ {φ(Z) + TN θ,η (S)⊥} ͱ͍͏ۭؒʹؚ·ΕΔ. ͨͩ͠, φ(Z) ҙͷ RAL ਪఆྔͷӨڹؔͰ, TN θ,η (S)⊥ ηϛύϥϝτϦο Ϋۭؒͷަิۭؒ ఆཧ 2 ηϛύϥϝτϦοΫ༗ޮͳਪఆྔ, ͦͷӨڹ͕ؔҰҙʹ well-defined Ͱܾఆ͞ Ε,φefficient = φ(Z) − {φ(Z)|TN θ,η (S)⊥} ͷཁૉ. ͪͳΈʹ, (h|U) projection of h ∈ H(ੵΛಋೖͨ͠ώϧϕϧτۭؒ) onto the space U (ઢܗۭؒ)
GEE ʹ͍ͭͯͷ Remarks Liang-Zeger ͷ GEE ͷηϛύϥϝτϦοΫϞσϧ (੍ϞʔϝϯτϞσϧ: 1 ࣍ͱ
2 ࣍ͷϞʔϝϯτʹ੍͚ͩΛஔ͍ͨϞσϧ) ҎԼͷಛΛͭ. ہॴ (ۙ༗) ޮਪఆྔ: ࢄؔͷԾఆ͕ਖ਼͚͠Ε, ༗ޮਪఆྔ Robustness: ແݶ࣍ݩͷύϥϝʔλਪఆ͕ඞཁ͕ͩ, ࢄؔΛ misspecify ͨ͠ͱͯ͠Ұகੑͱۙਖ਼نੑอ࣋ GEE ͷຊΛಡΊΘ͔Δ͚Ͳ, Working covariance matrix Λؒҧ͑ͯ ༗ޮੑࣦΘΕΔ͕, ͦͷଞͷ·͍͠ੑ࣭ (ۙਖ਼نੑͱҰகੑ) อ࣋Ͱ͖Δͬͯ͜ͱ